In order to find the place for the support of the table, we need to find the balancing point. In a triangle, the balancing point is a centroid of the triangle. Thus, we need to locate the centroid of the triangle. First, using the given coordinates, we will draw the triangle on a coordinate plane.

Let's recall that a centroid of a triangle is point of intersection of the triangle medians. Thus, to locate the centroid, we can draw the medians of our triangle and find their point of concurrency. The other way to find the centroid is to use the Centroid Theorem. Let's review what it states.

The medians of a triangle inersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side .

First, we can draw a median from the vertex with the coordinates

(3, 6) .

To do this, we need to know the coordinates of the midpoint of the opposite side. Let's find them substituting

(7, 10)

and

(5, 2)

into the Midpoint Formula.

M (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

SubstitutePoints

Substitute $(7, 10)$ & $(5, 2)$

M (\frac{7 + 5}{2}, \frac{1 0 + 2}{2})

AddTerms

Add terms

M (\frac{1 2}{2}, \frac{1 2}{2})

CalcQuot

Calculate quotient

M (6, 6)

Now, we can plot

M (6, 6)

on the coordinate plane and draw the median. Let's also name the vertices of the triangle for the following explanation to be easier.

According to the above theorem, the centroid of

△ A B C

is two thirds of the distance from

A

to

M .

If we call the centroid

D,

the following equality is true.

A D = \frac{2}{3} A M

To find

A M,

let's analyze the diagram. We can see that

A (3, 6)

and

M (6, 6)

have the same

y -

coordinate,

6 .

Hence, to find the measure of

\overline{A M},

we can calculate the difference between their

x -

coordinates.

A M = 6 - 3 = 3

Now, let's substitute

3

for

A M

into our equation and solve it for

A D .

A D = \frac{2}{3} A M

Substitute

$A M = 3$

A D = \frac{2}{3} (3)

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

A D = \frac{2 \cdot 3}{3}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

A D = 2

Since the measure of

A D

is

2,

by adding

2

to the

x -

coordinate of

A (3, 6)

we can find the coordinates of the centroid

D .

D (3 + 2, 6) \Rightarrow D (5, 6)

Therefore, the coordinates of the centroid of the triangle are

(5, 6) .

Exercise